You are quite right, the “properties” (in this case, the property “information” in some loose sense) of the “message” particle of Alice gets transported via the entangled state to Bob. As you emphasize, in that process the entanglement of the state shared between Alice and Bob gets transferred to entanglement between the two particles Alice has.
The deep reason behind why this is needed is the so-called “monogamy of entanglement”, which you can read more about here. Put simply, a particle can only be maximally entangled with one other particle, but not two. This is why the Alice’s particle initially entangled with Bob’s gets disentangled from it when entangled with the “message” particle.